Critically Damped System (ξ = 1)

Critically Damped System (ξ = 1)

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critically-damped-system-graph Fig.1(Critically Damped System)

  • Critically damped system(ξ=1): If the damping factor ξ is equal to one, or the damping coefficient c is equal to critical damping coefficient "cc", then the system is said to be a critically damped system.
ξ=1ORccc=1    c=cc\xi=1 \quad \text OR \quad {c \over c_c} = 1\implies c = c_c
  • Two roots for critically damped system are given by S1 and S2 as below:
S1=[ξ+ξ21]ωnS2=[ξξ21]ωnS_1 = \big [-\xi + \sqrt{\xi^2 -1} \big] \omega_n \\ S_2 = \big [-\xi - \sqrt{\xi^2 -1} \big] \omega_n
  • For ξ=1ξ=1; S1=S2=ωnS_1 = S_2 = -\omega_n
    Here both the roots are real and equal, so the solution to the differential equation can be given by
x=(A+Bt)eωnt...(1)x = (A + Bt)e^{-\omega_n t} \quad \quad ...(1)
  • Now differentiating equation (1) with respect to ‘t’, we get:
x˚=Beωntωn(A+Bt)eωnt...(2)\mathring x = Be^{-\omega_nt} - \omega_n(A + Bt)e^{-\omega_nt} \quad \quad ...(2)
  • Now, let at
    t=0t = 0 : x=X0x = X_0
    t=0t = 0 : x˚=0\mathring x =0

  • Substituting these values in equation (1):

X0=A...(3)X_0 = A \quad ...(3)
  • Same way, from equation (2), we get
0=Bωn(A+0)0=BωnAB=ωnAB=ωnX0...(4)\begin{aligned} 0&= B - \omega_n(A + 0) \\ 0&= B - \omega_nA \\ B&= \omega_nA \\ B&= \omega_nX_0 \qquad ...(4) \end{aligned}
  • Now putting the values of A and B in equation (1), we get:
x=(X0+ωnX0t)eωntx=X0(1+ωnt)eωnt...(5)\begin{aligned} x&= (X_0 + \omega_nX_0t)e^{-\omega_nt} \\ x&= X_0(1 + \omega_nt)e^{-\omega_nt} \qquad ...(5) \end{aligned}
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Conclusion

  • From above equation (5), it is seen that as time t increases, the displacement x decreases exponentially.

  • The motion of a critically damped system is aperiodic (aperiodic motion motions are those motions in which the motion does not repeat after a regular interval of time i.e non periodic motion) and so the system does not shows vibrations.

  • For critically damped systems, if a system is displaced from its initial position, it will try to reach its mean position in a very short time.

  • Critically damped systems are generally seen in hydraulic doors closer as it is necessary for the door to come to its initial position in a very short time.

critically-damped-system-graph Fig.1(Critically Damped System)

  • Critically damped system(ξ=1): If the damping factor ξ is equal to one, or the damping coefficient c is equal to critical damping coefficient "cc", then the system is said to be a critically damped system.
ξ=1ORccc=1    c=cc\xi=1 \quad \text OR \quad {c \over c_c} = 1\implies c = c_c
  • Two roots for critically damped system are given by S1 and S2 as below:
S1=[ξ+ξ21]ωnS2=[ξξ21]ωnS_1 = \big [-\xi + \sqrt{\xi^2 -1} \big] \omega_n \\ S_2 = \big [-\xi - \sqrt{\xi^2 -1} \big] \omega_n
  • For ξ=1ξ=1; S1=S2=ωnS_1 = S_2 = -\omega_n
    Here both the roots are real and equal, so the solution to the differential equation can be given by
x=(A+Bt)eωnt...(1)x = (A + Bt)e^{-\omega_n t} \quad \quad ...(1)
  • Now differentiating equation (1) with respect to ‘t’, we get:
x˚=Beωntωn(A+Bt)eωnt...(2)\mathring x = Be^{-\omega_nt} - \omega_n(A + Bt)e^{-\omega_nt} \quad \quad ...(2)
  • Now, let at
    t=0t = 0 : x=X0x = X_0
    t=0t = 0 : x˚=0\mathring x =0

  • Substituting these values in equation (1):

X0=A...(3)X_0 = A \quad ...(3)
  • Same way, from equation (2), we get
0=Bωn(A+0)0=BωnAB=ωnAB=ωnX0...(4)\begin{aligned} 0&= B - \omega_n(A + 0) \\ 0&= B - \omega_nA \\ B&= \omega_nA \\ B&= \omega_nX_0 \qquad ...(4) \end{aligned}
  • Now putting the values of A and B in equation (1), we get:
x=(X0+ωnX0t)eωntx=X0(1+ωnt)eωnt...(5)\begin{aligned} x&= (X_0 + \omega_nX_0t)e^{-\omega_nt} \\ x&= X_0(1 + \omega_nt)e^{-\omega_nt} \qquad ...(5) \end{aligned}
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Conclusion

  • From above equation (5), it is seen that as time t increases, the displacement x decreases exponentially.

  • The motion of a critically damped system is aperiodic (aperiodic motion motions are those motions in which the motion does not repeat after a regular interval of time i.e non periodic motion) and so the system does not shows vibrations.

  • For critically damped systems, if a system is displaced from its initial position, it will try to reach its mean position in a very short time.

  • Critically damped systems are generally seen in hydraulic doors closer as it is necessary for the door to come to its initial position in a very short time.

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